In spatstat and for calculating the maximum absolute difference
between the empirical Ripley's K-function (\hat(K), isotropic edge
correction) and the theoretical K-function for a simulated Poisson
process in a fixed window (suppose 3D case and fix number of points
(n) in W), we usually use a sequence of regular grid points. On the
other hand we know that the empirical K function is a step function
and hence the max occurs only at a jump point. suppose x is a typical
jump point (x is one of considered grid points), we have actually two
differences at x because we should also consider the difference
between \hat{K}(x-) and K(x-), where x- is the value just
infinitesimally smaller than x. The problem is if we calculate the
absolute difference only at the regular jump points like x, we shall
underestimate the true unknown maximum absolute difference between
\hat(K) and the theoretical K. How we can deal with this problem.
Shall taking very fine grids points would solve the problem? If yes
some word of theoretical reasoning please in direction of
programming.